The ratio of the incomes of P and Q is `5:4.` The ratio of their expenditure is `4:3` The savings of P more then that of Q by `16(2)/(3)%.` What percentage of his income does P spend ?

Options

A. `52(2)/(3)%`
B. `53(1)/(3)%`
C. `54(1)/(3)%`
D. `51(2)/(3)%`

Explanation

To solve the problem step by step, we will follow the information given in the question and use algebra to find the percentage of income that P spends.

### Step-by-Step Solution:

1. **Define the Incomes and Expenditures:**
- Let the incomes of P and Q be represented as:
- Income of P = \( 5x \)
- Income of Q = \( 4x \)
- Let the expenditures of P and Q be represented as:
- Expenditure of P = \( 4y \)
- Expenditure of Q = \( 3y \)

2. **Calculate Savings:**
- Savings of P = Income of P - Expenditure of P
\[ \text{Savings of P} = 5x - 4y \]
- Savings of Q = Income of Q - Expenditure of Q
\[ \text{Savings of Q} = 4x - 3y \]

3. **Set Up the Savings Equation:**
- According to the problem, the savings of P is more than the savings of Q by \( 16 \frac{2}{3} \% \).
- Convert \( 16 \frac{2}{3} \% \) to a fraction:
\[ 16 \frac{2}{3} \% = \frac{50}{3} \% \]
- This means:
\[ \text{Savings of P} = \text{Savings of Q} + \frac{50}{3} \% \text{ of Savings of Q} \]
- We can express this as:
\[ 5x - 4y = (4x - 3y) + \frac{50}{300} \times (4x - 3y) \]
- Simplifying gives:
\[ 5x - 4y = (4x - 3y) \left(1 + \frac{1}{6}\right) \]
- This simplifies to:
\[ 5x - 4y = \frac{7}{6}(4x - 3y) \]

4. **Clear the Fraction:**
- Multiply through by 6 to eliminate the fraction:
\[ 6(5x - 4y) = 7(4x - 3y) \]
- Expanding both sides:
\[ 30x - 24y = 28x - 21y \]

5. **Rearranging the Equation:**
- Rearranging gives:
\[ 30x - 28x = 24y - 21y \]
- Simplifying yields:
\[ 2x = 3y \quad \Rightarrow \quad \frac{y}{x} = \frac{2}{3} \]

6. **Find the Expenditure of P:**
- The expenditure of P is:
\[ \text{Expenditure of P} = 4y = 4 \left(\frac{2}{3}x\right) = \frac{8}{3}x \]

7. **Calculate the Percentage of Income that P Spends:**
- The percentage of income that P spends is given by:
\[ \text{Percentage spent} = \left(\frac{\text{Expenditure of P}}{\text{Income of P}}\right) \times 100 \]
- Substituting the values:
\[ \text{Percentage spent} = \left(\frac{\frac{8}{3}x}{5x}\right) \times 100 = \left(\frac{8}{15}\right) \times 100 \]
- Simplifying gives:
\[ \text{Percentage spent} = \frac{800}{15} = 53 \frac{1}{3} \% \]

### Final Answer:
P spends \( 53 \frac{1}{3} \% \) of his income.